Note on fractional sum of the divisor function

Source abstract

Let $d(n)$ be the divisor function and denote by $[t]$ the integral part of the real number $t$. We prove that $$\begin{align*} \sum_{n\leq x^{1/c}}d\left(\left[\frac{x}{n^c}\right]\right)=d_cx^{1/c}+O_{\varepsilon,c}(x^{θ_c+\varepsilon}), \end{align*}$$ where $d_c$ is a suitable constant, $$\begin{align*} θ_c=\frac{2(c+1)}{2c^2+(5+i)c+2}, \quad \text{for}\ 2κ_{i+1}/(1-κ_{i+1}+λ_{i+1})\leq c<2κ_i/(1-κ_i+λ_i), \end{align*}$$ the pair $(κ_i,λ_i)$ satisfies $$\begin{align*} (κ_1,λ_1)=\left(\frac{1}{14},\frac{11}{14}\right),\quad (κ_{i+1},λ_{i+1})=\left(\frac{κ_i}{2κ_i+2},\frac{κ_i+λ_i+1}{2κ_i+2}\right), \quad i\geq 1. \end{align*}$$ This result constitutes an improvement upon that of L. Wu (2025).